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General information group 11 up to 19
Magic squares of group 11-19 are constructed by means of combining a grid consisting of H- and/or
K-quadrants (each quadrant contains 2 times 8 digits), and a grid consisting of A-,B- and C-quadrants
(each quadrant contains 4 times 4 digits). Half of the amount of homogeneous H-grids (HHHH-grids)
can be matched with homogeneous A- and with homogeneous C-grids, the other half can be matched
with mixed AC-grids. Analogously half of the homogeneous K-grids can be matched with homogeneous
B- and with homogeneous C-grids, the other half can be matched with mixed BC-grids. Analogously half
of the mixed HK- or KH-grids can be matched with mixed AC- and BC-grids, the other half can be matched
with mixed ACC*B- or CABC*-grids.
Illustration group 11
Magic squares of group 11 are constructed by means of combining H-grids with A-grids.
In the example below a row grid with H4 in all four quadrants has been chosen.
H4 (row grid)
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Now the construction of the matching column grid. Fill the top left quadrant after A1, A2, or A3.
In the example A1 has been chosen. (Verify that A1*, a2* and a3* do not work!).
A1 (column grid), step 1
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In the 5th row only 1-6-7-0 is possible. And now you will find out that the third quadrant can only be
completed with an A-structure.
A1 (column grid), step 2
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The right half of the row grid must be filled with the digits 2, 3, 4, and 5. In column 5 it is only possible
to fill in 2-5-3-4, or 4-3-5-2 . In the example 4-3-5-2 has been chosen. With both options you can only
finish the upper right quadrant successfully when maintaining the A-structure. The down right quadrant
follows automatically, and has necessarily also the A-structure.
A1 (column grid), step 3
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A1 (column grid), step 4
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In total there are 3 (A1, A2, A3) x 2 (options of step 3) = 6 different column grids.
Finally you can combine row and column grid to produce the magic square. The square below contains
the X magic property (shown in blue).
1x digit from row grid +1 + 8x digit from column grid = most perfect 8x8 magic square
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The total amount of squares of group 11 is: 48 (row grids H) x 6 (column grids) x 2 (swapping row and
column grids) = 576; 24 x 6 = 144 of these squares have the extra magic property X.
Illustration group 12
Magic squares of group 12 are constructed by means of combining H-grids with C*-grids .
In the example below the same row grid as above (H4 repeated in all four quadrants) has been chosen.
H4 (row grid)
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Now you put C1*, C3* or C5* in the upper left corner (verify that C2*, C4* and C6* will not work in giving
a matching column grid). In theexample C1* has been chosen.
C1* (column grid), step 1
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In the 5th row you have only the matching option 1-6-7-0. And you only can finish the third quadrant success-
fully when maintaining the C*-structure.
C1* (column grid), step 2
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The right half of the row grid must be filled in with the digits 2, 3, 4, and 5. Column 5 needs 2-4-3-5 or 4-2-5-3.
In the example 2-4-3-5 has been chosen. With both options you can successfully finish the upper right quadrant
only when continuing the C*-structure. The down right quadrant follows automatically, and has necessarily the
C*-structure.
C1* (column grid), step 3
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C1* (column grid), step 4
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In total there are 3 (C1*, C3* or C5*) x 2 (options of step 3) = 6 different column grids.
Finally you can combine row and column grid to produce the magic square.
1x digit from row grid +1 + 8x digit from column grid = most perfect 8x8 magic square
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52
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15
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2
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61
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36
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31
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18
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45
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The total number of squares of group 12 is: 48 (row grids H) x 6 (column grids) x 2 (swapping row and
column grids) = 576; none of these squares can have the extra magic property X.
Illustration group 13
Magic squares of group 13 are constructed by means of combining H-grids with AC- or CA-grids.
Arbitrary we have constructed the following row grid:
H4 (row grid)
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Now we must find a matching AC column grid. We start filling the top left quadrant after A1.
A1 (column grid), step 1
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The 5th row needs necessarily 1-6-7-0. And you will find out that you can only finish the third quadrant
successfully when following the C* structure (if you follow the A-structure you get double pairings when
composing the final magic square):
A1 (column grid), step 2
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6
|
|
|
|
|
|
0
|
7
|
6
|
1
|
|
|
|
|
|
6
|
1
|
0
|
7
|
|
|
|
|
Column 5 needs 2-5-3-4 or 4-3-5-2. In the example 2-5-3-4 has been chosen. With both options you can
finish the upper right quadrant only successfully when following the A-structure. The down right quadrant
follows automatically, and has necessarily the C*-structure.
A1 (column grid), step 3
| 
